Advances in the Modelling of Motorcycle Dynamics - ResearchGate

274 R.S. SHARP ET AL.These checks on any equilibrium state are substantially independent of the fullequations of motion on which the simulation model depends and it is reassuringthat they are satisfied.8. Typical ResultsThe main uses of a model such as that described are (a) general simulation of responsesto defined steering control inputs, possibly involving hardware in the loop(b) determination of steady-state equilibrium cornering “trim” states (c) linearizationof the equations to represent small motions in the neighbourhood of a trimstate (d) root locus calculations for constant lean angle and varying speed or viceversaand (e) frequency response calculations to find gains and phases in sustainedmotion involving sinusoidal forcing from the steering system or from road undulations[30]. The power computations also allow determination in detail of wherethe engine power is dissipated in steady turning.Trim state determination is a necessary forerunner to stability and frequencyresponse computations, to enable the linearisation to be done correctly. Also, asdescribed in Section 6, speed and lean angle controllers are likely to be needed toallow the trim states to be found, over a reasonably full range of feasible speeds andlean angles. Some of these uses and some behavioural properties of the machine infocus are illustrated next.The model was first used to simulate a straight line run from 1 to 75 m/s with avery small constant acceleration of 0.05 m/s 2 . This gives the trim state, changingwith the speed, from which small perturbations are considered to occur and forwhich a linearised model is appropriate. The linearised model, having a free steeringsystem with the feedback steering controller disabled, was then used to obtain theroot locus plot shown in Figure 18. The machine, as represented, is stable for straightrunning throughout the speed range above about 6 m/s. Also shown in Figure 18are the loci for the nominal motorcycle but with the rear frame torsional stiffnessdivided by 2 and then 4 with the frame twist damping coefficient reduced by factorsof 0.7071 and 0.5 respectively.The high speed weave stability is compromised significantly by the reduction instiffness and the wobble problem is transferred from high speed to medium speedby these changes. This aligns with earlier findings, that flexible frames promotemedium speed wobble, while very stiff frames give more of a potential problemat high speeds, implying the need for a steering damper to ensure adequatemargins.The damping coefficient associated with the rider upper body lean freedom isnow varied, with root loci being shown in Figure 19. Rider damping can be seento influence the weave mode only where the damping is plentiful but it contributesusefully to the stability of the wobble mode at high speed. The results are consistentwith the idea that lighter riders are more likely to suffer wobble oscillations thanheavier ones, in accord with anecdotal evidence.